Controlled jump in the Clifford hierarchy

TL;DR

This paper introduces a systematic method to elevate Clifford operations to higher levels of the hierarchy through controlled gates, providing theoretical bounds and explicit constructions, with applications in quantum gate synthesis.

Contribution

It establishes a sharp controlled-jump rule in the Clifford hierarchy, quantifies resource requirements for high-level jumps, and proposes a protocol for logical phase gate implementation.

Findings

Controlled gates increase hierarchy level based on Pauli periodicity.

High-level jumps require exponentially many target qubits.

Explicit families of Clifford operations achieve near-optimal jumps.

Abstract

We develop a simple and systematic route to higher levels of the qubit Clifford hierarchy by coherently controlling Clifford operations. Our approach is based on Pauli periodicity, defined for a Clifford unitary $U$ as the smallest integer $m\ge 1$ such that $U^{2^{m}}$ is a Pauli operator up to phase. We prove a sharp controlled-jump rule showing that the controlled gate $CU$ lies strictly in level $m+2$ of the hierarchy, and equivalently that $CU$ lies in level $k$ if $U^{2^{k-2}}$ is Pauli while no smaller positive power of $U$ is Pauli. We further quantify the resources required to realize large level jumps in the Clifford hierarchy by proving an essentially tight upper bound on Pauli periodicity as a function of the number of qubits, which implies that accessing high hierarchy levels through controlled Cliffords requires a number of target qubits that grows exponentially with the desired level. We complement this limitation with explicit infinite families of Pauli-periodic Cliffords whose controlled versions achieve asymptotically optimal jumps. As an application, we propose a protocol for preparing logical catalyst states that enable logical $Z^{1/2^k}$ phase gates via phase kickback from a single jumped Clifford.